Date post: | 17-Dec-2015 |
Category: |
Documents |
Upload: | jacob-riley |
View: | 213 times |
Download: | 0 times |
Big Words, Busy Beavers, andThe Cake of Computing
David Evanswww.cs.virginia.edu/evans
flickr: mwichary
TAPESTRY WorkshopUniversity of Virginia15 July 2009
flickr: chotda
graphicsgames
web appsGUIsJava
Frosting: Doing Cool Stuff with Computers
flickr: taryn
flickr: sukey2
flickr: jcsupersmith
NCAA DefinitionThe current core-curriculum areas were legislated in the early 1980's. At that time, computer-science courses were programming based and academic in nature. In today's secondary education environment, the vast majority of computer courses no longer contain programming elements but teach life skills, such as the use of a desktop computer and software applications. Although these software and keyboarding skills may be beneficial to college-bound students, they are not academic in nature. … It should be noted that computer courses that include a significant element of programming might be encompassed in the mathematics-curriculum requirement.
Revision of NCAA Eligibility Requirements, August 2005
flickr: chotda
graphicsgames
web appsGUIsJava
Recursion
Universality
Abstraction
Recursive Definitions
What’s the l o n g e s t
word in the English
language?
Longest Words?honorificabilitudinitatibus (27 letters, longest by Shakespeare)
With honor.antidisestablishmentarianism (28 letters)
Movement against division of church and state.hippopotomonstrosesquipedaliophobia (35 letters)
Fear of long words.pneumonoultramicroscopicsilicovolcanoconiosis (45 letters)
(longest word in most dictionaries)Lung disease contracted from volcanic particles.
Like all words, these words are “made up”.
Making Longer Wordsantihippopotomonstrosesquipedaliophobia
Against the fear of long words.
antiantihippopotomonstrosesquipedaliophobiaAgainst a thing against the fear of long
words.
Language is Recursive No matter what word you think is the longest word, I can always make up a longer one!
word ::= anti-word
By itself, this definition of word is circular.
Zero, One, Infinity
word ::= anti-word
word ::= hippopotomonstrosesquipedaliophobia
This rule can make 0 words.
This rule can make 1 word.
word ::= anti-wordword ::= hippopotomonstrosesquipedaliophobia
These two rules can make infinitely many words, enough to express all ideas in the universe!
Recursive Definitions Everywhere
• Language– Words, Sentences, Structures
• Nature– Plant Growth, Quantum Physics, DNA
• Mathematics– Numbers, Arithmetic Algorithms
• Music– Harmony, structure
• Computing– Data, procedures
Red Hot and Blue by Paul DiOrio, Rachel PhillipsWes Weimer may talk more about this tomorrow!
flickr: chotda
graphicsgames
web appsGUIsJava
Recursion
Universality
Abstraction
Biggest Number Game
• When I say “GO”, write down the biggest number you can in 30 seconds.
• Requirement: – Must be an exact number– Must be defined mathematically
• Biggest number wins!
Countdown Clock
3029282726252423222120191817161514131211109876543210STOP
What’s so special about computers?
Apollo Guidance Computer (1969)
Colossus (1944)
Cray-1 (1976)
Palm Pre (2009)Flickr: louisvolantApple II (1977)
Honeywell Kitchen Computer (1969)
Toaster Science?
“Computers” before WWII
Mechanical Computing
Modeling Computers• Input– Without it, we can’t describe a problem
• Output– Without it, we can’t get an answer
• Processing– Need some way of getting from the input to the
output• Memory– Need to keep track of what we are doing
Modeling Input
Engelbart’s mouse and keypad
Punch Cards
Altair BASIC Paper Tape, 1976
Turing’s Model
“Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book.”
Alan Turing, On computable numbers, with an application to the Entscheidungsproblem, 1936
Modeling Pencil and Paper
# C S S A 7 2 3
How long should the tape be?
... ...
Modeling Output
• Blinking lights are cool, but hard to model
• Use the tape: output is what is written on the tape at the end
Connection Machine CM-5, 1993
Modeling Processing (Brains)
Look at the current state of the computation
Follow simple rules about what to do next
Modeling Processing• Evaluation Rules– Given an input on our tape, how do we evaluate
to produce the output• What do we need:– Read what is on the tape at the current square– Move the tape one square in either direction– Write into the current square
0 0 1 1 0 0 1 0 0 0
Is that enough to model a computer?
Modeling Processing (Brains)
• Follow simple rules• Remember what you
are doing
“For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.”
Alan Turing
Turing’s Model: Turing Machine
1
Start
2
Input: #Write: #Move:
# 1 0 1 1 0 1 1... ...1 0 1 1 0 1 1 1 #
Input: 1Write: 0Move:
Input: 1Write: 1Move:
Input: 0Write: 0Move: 3
Input: 0Write: #Move: Halt
Infinite Tape: Finite set of symbols, one in each square Can read/write one square each step
Controller:Limited (finite) number of states
Follow rules based on current state and read symbol
Write one square each step, move left or right or halt, change state
What makes a good model?
Copernicus
F = GM1M2 / R2
Newton Ptolomy
Questions about Turing’s Model
• How well does it match “real” computers?– Can it do everything they can do?– Can they do everything it can do?
• Does it help us understand and reason about computing?
Church-Turing Thesis• All mechanical computers are equally powerful*
• There exists some Turing machine that can simulate any mechanical computer
• Any computer that is powerful enough to simulate a Turing machine, can simulate any mechanical computer
*Except for practical limits like memory size, time, display, energy, etc.
Power of Turing Machine
• Can it add?
• Can it carry out any computation?
• Can it solve any problem?
Performing Addition
• Input: a two sequences of digits, separated by + with # at end.
e.g., # 1 2 9 3 5 2 + 6 3 5 9 4 #• Output: sum of the two numbers
e.g., # 1 9 2 9 4 6 #
Addition ProgramFind the rightmost digit of the first number:
A: look for +Start
+, +, L
B: read last digit
0, 0, R
1, 1, R
9, 9, R...
Read Write Move
C00, X, R
C99, X, R
C11, X, R
...
Addition, ContinuedFind the rightmost digit of the second number:
C4
4, X, R
Must duplicate this for each first digit – states keep track of first digit!
look for #
1, 1, RX, X, R ...
#, #, RD4: read last digit
E40, X, R
E7
3, X, R
6, X, R E10
......
Power of Turing Machine
Can it add?
• Can it carry out any computation?
• Can it solve any problem?
Universal Machine
Description of a Turing Machine M Input
UniversalMachine
Result tape of running M on Input
A Universal Turing Machine can simulateany Turing Machine running on any Input!
Manchester Illuminated Universal Turing Machine, #9 from http://www.verostko.com/manchester/manchester.html
Universal Computing Machine
2-state, 3-symbol Turing machine proved universal by Alex Smith in 2007
What This Means• Your cell phone, watch, iPod, etc. has a
processor powerful enough to simulate a Turing machine
• A Turing machine can simulate the world’s most powerful supercomputer
• Thus, your cell phone can simulate the world’s most powerful supercomputer (it’ll just take a lot longer and will run out of memory)
Are there problems computers can’t solve?
The “Busy Beaver” Game
• Design a Turing Machine that:– Uses two symbols (e.g., “0” and “1”)– Starts with a tape of all “0”s– Eventually halts (can’t run forever)– Has N states
• Goal: machine runs for as many steps as possible before eventually halting
Tibor Radó, 1962
Busy Beaver: N = 1
AStart
Input: 0Write: 1Move: Halt
0 0 0 0 0 0 0 0... ...
0 0 0 0 0 0 0 0 0
H
BB(1) = 1 Most steps a 1-state machine that halts can make
AStart
B
Input: 0Write: 1Move:
0 0 0 0 0 0 0 0... ...
0 0 0 0 0 0 0 0 0
Input: 0Write: 1Move:
H
Input: 1Write: 1Move: Halt
Input: 1Write: 1Move:
BB(2) = 6
AStart
BC
D E
F
0/1/R
1/0/L
0/0/R
1/0/R
0/1/L 1/1/R
0/0/L
1/0/L
0/0/R
1/1/R
0/1/L
H
1/1/H
6-state machine found by Buntrock and Marxen, 2001
AStart
BC
DE
F
0/1/R
1/0/L
0/0/R
1/0/R
0/1/L 1/1/R
0/0/L
1/0/L
0/0/R
1/1/R
0/1/L
H
1/1/H
300232771652356282895510301834134018514775433724675250037338180173521424076038326588191208297820287669898401786071345848280422383492822716051848585583668153797251438618561730209415487685570078538658757304857487222040030769844045098871367087615079138311034353164641077919209890837164477363289374225531955126023251172259034570155087303683654630874155990822516129938425830691378607273670708190160525534077040039226593073997923170154775358629850421712513378527086223112680677973751790032937578520017666792246839908855920362933767744760870128446883455477806316491601855784426860769027944542798006152693167452821336689917460886106486574189015401194034857577718253065541632656334314242325592486700118506716581303423271748965426160409797173073716688827281435904639445605928175254048321109306002474658968108793381912381812336227992839930833085933478853176574702776062858289156568392295963586263654139383856764728051394965554409688456578122743296319960808368094536421039149584946758006509160985701328997026301708760235500239598119410592142621669614552827244429217416465494363891697113965316892660611709290048580677566178715752354594049016719278069832866522332923541370293059667996001319376698551683848851474625152094567110615451986839894490885687082244978774551453204358588661593979763935102896523295803940023673203101744986550732496850436999753711343067328676158146269292723375662015612826924105454849658410961574031211440611088975349899156714888681952366018086246687712098553077054825367434062671756760070388922117434932633444773138783714023735898712790278288377198260380065105075792925239453450622999208297579584893448886278127629044163292251815410053522246084552761513383934623129083266949377380950466643121689746511996847681275076313206
(1730 digits)
Best found before 2001, only 925 digits!
In Dec 2007, Terry and Shawn Ligocki beat this: 2879 digits!
Busy Beaver NumbersBB(1) = 1BB(2) = 6BB(3) = 21BB(4) = 107BB(5) = Unknown!
Best so far is 47,176,870
BB(6) > 102879 Discovered 2007
Winning the “Biggest number” game: BB(BB(BB(BB(111111111))))
flickr: climbnh2003
Computing Busy Beaver Numbers
• Input: N (number of states)• Output: BB(N)– The maximum number of steps a Turing Machine
with N states can take before halting
Is it possible to design a Turing Machine that solves the Busy Beaver Problem?
The Halting Problem
• Input: a description of a Turing Machine• Output: “1” if it eventually halts, “0” if it never
halts, starting on a tape full of “0”s.
Is it possible to design a Turing Machine that solves the Halting Problem?
“Solves” means for all inputs, the machine finishes and produces the right answer.
Example
AStart
B
0/0/R
H
1/1/H
0/0/L
0 (it never halts)
HaltingProblem
Solver
HaltingProblem
Solver
HaltingProblem
Solver
Example
Impossibility Proof!
HaltingProblem
Solver
HXY
HaltingProblem
SolverF
0, 0, H1, 0, R
*, *, L
*, *, R
Impossible to make Halting Problem Solver
• If it outputs “0” on the input, the input machine would halt (so “0” cannot be correct)
• If it outputs “1” on the input, the input machine never halts (so “1” cannot be correct)
If it halts, it doesn’t halt! If it doesn’t halt, it halts!
Busy Beaver is Impossible Too!
• If you could solve it, could solve Halting Problem:– Input machine has N states– Compute BB(N)– Simulate input machine for BB(N) steps– If it ever halts, it must halt by now
• ... but we know that is impossible, so it must be impossible to computer BB(N)
The BB numbers are so big you can’t even compute them!
Recap• A computer is something that can carry out
well-defined steps:– Read and write on scratch paper, follow rules, keep
track of state• All computers are equally powerful– If a machine can simulate any step of another
machine, it can simulate the other machine (except for physical limits)
– What matters is the program that defines the steps
You can have your frosting and eat cake too!
Questions
David [email protected]
http://www.computingbook.org/
Some Sources:Matthias Felleisen, Shriram Krishnamurthi , Why Computer Science Doesn't Matter, Communications of the ACM July 2009.
Scott Aaronson, Who Can Name the Bigger Number?, http://www.scottaaronson.com/writings/bignumbers.html